5+55+555...n terms. Change this term into G.P. then find the sum of this series?
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Sum = 5 + 55 + 555 + …. n terms.
= 5/9[9 + 99 + 999 + …. n terms]
= 5/9[(10 – 1) + (100 – 1) + (1000 – 1) + … n terms]
= 5/9[10 + 100 + 1000 ….. – (1 + 1 + … 1)]
= 5/9[10(10n – 1)/(10 – 1) + (1 + 1 + … n times))
= 50/81(10n – 1) – 5n/9
= 5/9[9 + 99 + 999 + …. n terms]
= 5/9[(10 – 1) + (100 – 1) + (1000 – 1) + … n terms]
= 5/9[10 + 100 + 1000 ….. – (1 + 1 + … 1)]
= 5/9[10(10n – 1)/(10 – 1) + (1 + 1 + … n times))
= 50/81(10n – 1) – 5n/9
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